vector spaces 4.5 basis and dimension1. v = span(s) and 2. s is linearly independent. in words, we...
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Vector Spaces
§4.5 Basis and Dimension
Satya Mandal, KU
October 23
Satya Mandal, KU Vector Spaces §4.5 Basis and Dimension
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Goals
Discuss two related important concepts:
◮ Define Basis of a Vectors Space V .
◮ Define Dimension dim(V ) of a Vectors Space V .
Satya Mandal, KU Vector Spaces §4.5 Basis and Dimension
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Basis
Let V be a vector space (over R). A set S of vectors in V iscalled a basis of V if
1. V = Span(S) and
2. S is linearly independent.
◮ In words, we say that S is a basis of V if S in linealry
independent and if S spans V .
◮ First note, it would need a proof (i.e. it is a theorem)that any vector space has a basis.
Satya Mandal, KU Vector Spaces §4.5 Basis and Dimension
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Continued
◮ The definition of basis does not require that S is a finiteset.
◮ However, we will only deal with situations whenS = {v1, v2, . . . , vn} is a finite set.
◮ If V has a finite basis S = {v1, v2, . . . , vn}, then we saythat V is finite dimensional. Otherwise, we say that V isinfinite dimensional.
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Examples from the Textbook
◮ Reading Assignment: §4.5 Example 1-5.◮ Example 1. The set S = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is
a basis of the 3−space R3.
Proof.◮ Given any (x , y , z) ∈ R
3 we have
(x , y , z) = x(1, 0, 0) + y(0, 1, 0) + z(0, 0, 1).
So, any x , y , z) ∈ R3 is a linearl combinations of
elements in S . So, R3 = Span(S).◮ Also, S us linealry independent:
x(1, 0, 0)+y(0, 1, 0)+z(0, 0, 1) = (0, 0, 0) =⇒ x = y = z = 0.
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Example 1a.
Similarly, a basis of the n−space Rn is given by the set
S = {e1, e2, . . . , en}
where
e1 = (1, 0, . . . , 0), e2 = (0, 1, . . . , 0), . . . , en = (0, 0, . . . , 1).
This one is called the standard basis of Rn.
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Example 2 (edited)
The set S = {(1,−1, 0), (1, 1, 0), (1, 1, 1)} is a basis of R3.
Proof.◮ First we prove Span(S) = R
3. Let (x , y , z) ∈ R
3. We
need to find a, b, c such that
(x , y , z) = a(1,−1, 0) + b(1, 1, 0) + c(1, 1, 1)
So,
1 1 1−1 1 10 0 1
a
b
c
=
x
y
z
notationally Aa = v
Satya Mandal, KU Vector Spaces §4.5 Basis and Dimension
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Continued
Compute inverse of A:
[A|I3] =
1 1 1 1 0 0−1 1 1 0 1 00 0 1 0 0 1
Add first row to second
1 1 1 1 0 00 2 2 1 1 00 0 1 0 0 1
Subtract third row from first and subtract 2 times third rowfrom second:
1 1 0 1 0 −10 2 0 1 1 −20 0 1 0 0 1
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Continued
Multiply second row by .5; then subtract second row from first:
1 1 0 1 0 −10 1 0 .5 .5 −10 0 1 0 0 1
7→
1 0 0 .5 −.5 00 1 0 .5 .5 −10 0 1 0 0 1
So,
A−1 =
.5 −.5 0
.5 .5 −10 0 1
Satya Mandal, KU Vector Spaces §4.5 Basis and Dimension
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Continued
a
b
c
= A−1
x
y
z
=
.5 −.5 0
.5 .5 −10 0 1
x
y
z
Hence
(x , y , z) = a(1,−1, 0) + b(1, 1, 0) + c(1, 1, 1) ∈ Span(S).
Therefore, Span(S) = R3.
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◮ Now, we prove S is linearly independent. Let
a(1,−1, 0) + b(1, 1, 0) + c(1, 1, 1) = (0, 0, 0).
We need to prove a = b = c = 0. In fact, in the matrix
from, this equation is A
a
b
c
=
000
where A
is as above. Since, A is non-singular,
a
b
c
=
000
So,
S is linearly independent.
◮ Since, span(S) = R3 and S is linearly independent, S
forms a bais of R3.
Satya Mandal, KU Vector Spaces §4.5 Basis and Dimension
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Examples 4
◮ Let P3 be a vector space of all polynomials of degree lessof equal to 3. Then S{1, x , x2, x3} is a basis of P3.
Proof. Clearly span(S) = P3. Also S is linearlyindependent, because
a1 + bx + cx2 + dx3 =⇒ a = b = c = d = 0.
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Example 5.
◮ Let M3,2 be the vector space of all 3× 2 matrices. Let
A1,1 =
1 00 00 0
,A1,2 =
0 10 00 0
,A2,1 =
0 01 00 0
,
A2,2 =
0 00 10 0
,A3,1 =
0 00 01 0
,A3,2 =
0 00 00 1
Then,A = {A11,A12,A2,1,A2,2,A3,1,A3,2}
is a basis of M3,2.
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Theorem 4.9
Theorem 4.9(Uniqueness of basis representation): Let V bea vector space and S = {v1, v2, . . . , vn} be a basis of V .
Then, any vector v ∈ V can be written in one and only oneway as linear combination of vectors in S .
Proof. Suppose v ∈ V . Since Span(S) = V
v = a1v1 + a2v2 + · · ·+ anvn where ai ∈ R.
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Now suppose there are two ways:
v = a1v1+a2v2+· · ·+anvn and v = b1v1+b2v2+· · ·+bnvn
We will prove a1 = b1, a2 = b2, . . . , an = bn.
Subtracting 0 = (a1− b1)v1+(a2− b2)v2+ · · ·+(an− bn)vn
Since, S is linearly independent,a1 − b1 = 0, a2 − b2 = 0, . . . , an − bn = 0 ora1 = b1, a2 = b2, . . . , an = bn. The proof is complete.Reading Assignment: §4.5 Example 6
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Theorem 4.10
Theorem 4.10 (Bases and cardinalities) Let V be a vectorspace and S = {v1, v2, . . . , vn} be a basis of V , containing n
vectors. Then any set containing more than n vectors in V islinearly dependent.Proof.Let T = {u1,u2, . . . ,um} be set of m vectors in V
with m > n. For simplicity, assume n = 3 and m = 4. So,S = {v1, v2, v3} and T = {u1,u2,u3,u4}. To prove that T isdependent, we will have to find scalers x1, x2, x3, x4, not allzero, such that not all zero,
x1u1 + x2u2 + x3u3 + x4u4 = 0 Equation − I
Subsequently, we will show that Equation-I has non-trivialsolution.
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Continued
Since S is a basis we can write
u1 = c11v1 +c12v2 +c13v3u2 = c21v1 +c22v2 +c23v3u3 = c31v1 +c32v2 +c33v3u4 = c41v1 +c42v2 +c43v3
We substitute these in Equation-I and re-group:
(c11x1 +c21x2 +c31x3 +c41x4)v1+(c12x1 +c22x2 +c32x3 +c42x4)v2+(c13x1 +c23x2 +c33x3 +c43x4)v3 = 0
Since S = {v1, v2, v3} is linearly independent, the coeffients ofv1, v2, v3 are zero. So, we have (in the next frame):
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Continued
c11x1 +c21x2 +c31x3 +c41x4 = 0c12x1 +c22x2 +c32x3 +c42x4 = 0c13x1 +c23x2 +c33x3 +c43x4 = 0
This is a system of three homogeneous linear equations in fourvariables. (less equations than number of variable. So, thesystem has non-tirvial (infinitley many) solutions. So, thereare x1, x2, x3, x4, not all zero, so that Equation-I is valid. So,T = {u1,u2,u3,u4} is linealry dependent. The proof iscomplete.
Reading Assignment: §4.5 Example 7
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Theorem 4.11
Suppose V is a vector space. If V has a basis with n elementsthen all bases have n elements.Proof.Suppose S = {v1, v2, . . . , vn} andT = {u1,u2, . . . ,um} are two bases of V .
Since, the basisS has n elements, and T is linealryindependent, by the thoerem above m cannot be bigger thann. So, m ≤ n.
By switching the roles of S and T , we have n ≤ m. So,m = n. The proof is complete.
Reading Assignment: §4.5 Example 8.
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Dimension of Vactor Spaces
Definition. Let V be a vector space. Suppose V has a basisS = {v1, v2, . . . , vn} consisiting of n vectors. Then, we say n
is the dimension of V and write dim(V ) = n. If V consists ofthe zero vector only, then the dimension of V is defined to bezero.We have
◮ From above example dim(Rn) = n.
◮ From above example dim(P3) = 4. Similalry,dim(Pn) = n + 1.
◮ From above example dim(M3,2) = 6. Similarly,dim(Mn,m) = mn.
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Dimensions of Subspaces
◮ If W is a subspace of V , one can prove, then
dim(W ) ≤ dim(V ).
◮ Reading Assignment: §4.5 Example 9, 10, 11.
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Dimensions of Subspaces: Examples
◮ Example 9 (edited)◮
Let W = {(x , y , 2x + 3y) : x , y ∈ R}
The W is a subspace of R3 and dim(W ) = 2.Proof.Given (x , y , 2x + 3y) ∈ W , we have
(x , y , 2x + 3y) = x(1, 0, 2) + y(0, 1, 3)
This shows span({(1, 0, 2), (0, 1, 3)}) = W . Also{(1, 0, 2), (0, 1, 3)} is linearly independent. So,{(1, 0, 2), (0, 1, 3)} is a basis of W and dim(W ) = 2.
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Dimensions of Subspaces: Examples
Example 10 (edited) Let
S = {(1, 3,−2, 13), (−1, 2,−3, 12), (2, 1, 1, 1)}
and W = span(S). Then W is a subspace of R4 anddim(W ) = 2.
◮ Proof.Denote the three vectors in S by v1, v2, v3.◮ Then v3 = v1 − v2. Write T = {v1, v2}.◮ It follows, any linear combination of vectors in S is also a
linear combination of vectors in T .
◮
So, W = span(S) = span(T ).
◮ Also T is linearly indpendent. So, T is a basis anddim(W ) = 2.
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Theorem 4.12
Theorem 4.12. (Basis Tests): Let V be a vector space anddim(V ) = n.
◮ If S = {v1, v2, . . . , vn} is a linearly independent set in V
(consisting of n vectors), then S is a basis of V .
◮ If S = {v1, v2, . . . , vn} span V , then S is a basis of V
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Exercise
◮ Exercise 12 (edited). Let S = {(4,−3), (12,−9)}.Why S is not a basis for R2?Answer: S is linearly dependent. This is immediatebecause the first vector is a multiple of the second.
◮ Exercise 20 (edited). Why S is not a basis for R3 where
S = {(6, 4, 1), (3,−5, 1), (8, 13, 6), (0, 6, 9)}
Answer: Here dim(R3) = 3. So, any basis will have 3vectors, while S has four.
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Exercise
◮ Exercise 23 (edited). LetS = {1− x , 1− x2, 3x2 − 2x − 1}. Why S is not a basisfor P2?Answer: dimP2 = 3 and S has 3 elements. So, we haveto give different reason. In fact, S is linealry dependent:
3x2 − 2x − 1 = 2(1− x)− 3(1− x2)
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Exercise
◮ Exercise 28 (edited). Why S is not a basis for M22,where
S =
{[
1 00 1
]
,
[
1 01 1
]
,
[
1 10 1
]}
Answer: dim(M22) = 4 and S has 3 elements.
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Exercise
Exercise 28 (edited). Is S forms a basis for M22, where
S =
{[
1 00 1
]
,
[
1 01 1
]
,
[
1 10 1
]
,
[
1 11 0
]}
Answer: dim(M22) = 4 and S has 4 elements. Further, S islinearly independent. So, S is a basis of M22. To see they arelinearly independent: Let
a
[
1 00 1
]
+b
[
1 01 1
]
+c
[
1 10 1
]
+d
[
1 11 0
]
=
[
0 00 0
]
[
a + b + c + d c + d
b + d a + b + c
]
=
[
0 00 0
]
⇒ a = b = c = d = 0
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Homework
Homework: §4.5 Exercise 7, 8, 9, 10, 15, 16, 17,, 18, 26, 27,45, 51, 52, 56, 71, 72, 76.
Satya Mandal, KU Vector Spaces §4.5 Basis and Dimension